explanations of the term structure of interest rates

This PDF is a selection from an out-of-print volume from the NationalBureau of Economic Research

Volume Title: The Cyclical Behavior of the Term Structure of InterestRates

Volume Author/Editor: Reuben A. Kessel

Volume Publisher: NBER

Volume ISBN: 0-87014-405-7

Volume URL: http://www.nber.org/books/kess65-1

Publication Date: 1965

Chapter Title: Explanations of the Term Structure of Interest Rates

Chapter Author: Reuben A. Kessel

Chapter URL: http://www.nber.org/chapters/c1662

Chapter pages in book: (p. 5 - 43)

1

EXPLANATIONS

OF THE TERM STRUCTURE

OF INTEREST RATES

IT IS THE THESIS of this investigation that the term structure ofinterest rates can be explained better by a combination of theexpectations and liquidity preference hypotheses than by eitherhypothesis alone. Alternatively, these two hypotheses can be viewedas complementary explanations of the same phenomenon—the termstructure of interest rates. The evidence to be examined in supportof this view falls into two classes. One is the findings of previousinvestigators; the works of Macaulay, Culbertson, Meiselman,Walker, and Hickman contain evidence relevant for evaluating thesubstantive merits of this thesis. The other class consists of evidencegathered as part of the present investigation.

A. What Is the Expectations Hypothesis.

The expectations hypothesis has been enunciated by Fisher, Keynes,Hicks, Lutz, and others.1 It has had widespread appeal for theoreti-cal economists primarily as a result of its consistency with the waysimilar phenomena in other markets, particularly futures markets,are explained. In contrast, this hypothesis has been widely rejectedby empirically minded economists and practical men of affairs. Itwas rejected by economists because investigators have been unableto produce evidence of a relationship between the term structure ofinterest rates and expectations of future short-term rates. (Others

1 See Friedrich A. Lutz, "The Structure of Interest Rates," in the AmericanEconomic Association, Readings in the Theoiy of Income Distribution, 1'liila-deiphia, 1946, p. 499; and Joseph \V, Conard, 4n Introduction tO the Theory ofInterest, University of California Press, 1959, Part III.

6 Explanations of thehave found it difficult to accept the view that long- and short-termsecurities are perfect substitutes for one another in the market.)Meiselman contends that previous investigators have not devisedoperational implications of the expectations hypothesis. Moreover,he contends, they have examined propositions which were mis-takenly attributed to the expectations hypothesis, and when thesepropositions were found to be false, they rejected the expectationshypothesis.2

Briefly, the expectations hypothesis asserts that a long-term rateconstitutes an average (a weighted average in the case of coupon-bearing securities) of expected future short-term rates. It says thatforward rates (or marginal rates of interest) constitute unbiasedestimates of future spot rates.3 It is based on the assumption thatshort- and long-term securities, default risks aside, can be usefullyviewed as identical in all respects except maturity. It implies thatthe expected value of the returns derived from holding long- andshort-term securities for identical time periods are the same.

The word future should be emphasized in discussing the expec-tations hypothesis, since it concerns the effects of expectations aboutfuture short-term rates upon the current term structure of interestrates. To illustrate with a simplified example: assume that two-yearsecurities yield 3 per cent and one-year securities 2 per cent. Theforward rate on one-year money one year hence, or the marginalcost of extending a one-year term to maturity for an additionalyear, is 4 per cent; this is arithmetic, not the expectations hypoth-esis. The expectations hypothesis, as interpreted by Lutz and Meisel-man, but not by Hicks, states that the forward rates are unbiasedestimates of future short-term rates. For the preceding example,it implies that the market expects the rate on one-year securitiesone year hence to be 4 per cent. Four per cent is not only theforward rate—it is the expected one-year rate one year hence; i.e.,it is what the market thinks the one-year rate will be one yearhence.

2 David Meiselman, The Term Structure of Interest Rates, Englewood Cliffs,New Jersey, 1962, pp. 10 and 12.

A spot rate is a rate on funds for immediate delivery; it is today's rate formoney to be delivered today for a specified period of time. In contrast, a forwardrate is today's rate for money to be delivered in the future for a specified periodof time. This time period could be anything, a day, a year, or a decade,

Term Structure of Interest Rates 7

Conversely, assume a 2 per cent rate on two-year maturities anda 3 per cent rate on one-year maturities.. Then the yield on one-yearsecurities one year hence which will equalize the net yield fromholding two one-year securities successively with that of holding onetwo-year security is 1 per cent. This must follow if one accepts theview that securities are alike in all respects excep.t term to maturity.4

.B. Existing Evidence1. MACAULAY

Macaulay was among the first to produce empirical evidence thatrelated long-term rates to expectations of future short-term rates.Before the founding of the Federal Reserve System, there existed apronounced and well-known seasonal in the call money rate. Thewidespread knowledge of the existence of this seasonal implied thattime money rates, which are loans from one to six months that areotherwise similar to call money loans, should turn up before theseasonal rise in call money rates. Macaulay found that time moneyrates did in fact anticipate the seasonal rise in call money rates andconcluded that this constituted ". . . evidence of definite and rela-tively successful forecasting." Macaulay was unable to uncoveradditional evidence of successful forecasting. He warned againstconcluding that forecasting was not attempted. Macaulay's conten-tion was that evidence of successful forecasting is rare because suc-cessful forecasting is also rare.6

2. HICKMAN

W. Braddock Hickman, in a preliminary, unpublished, but never-theless widely cited and read, NBER manuscript prepared in 1942,reports the results of his tests of the expectations hypothesis.7 LikeMacaulay, he sought evidence of successful forecasting; unlikeMacaulay, he failed to find it. He compared observed or actualyield curves with those predicted one year 'or more ahead by the

4 These' calculations ignore compounding of interest and intermediate pay-ments in the form of coupons.

5 Frederick R. Macaulay, Movements of Interest Rates, p. 36. The reappear.ance of a seasonal in the money market in recent years has made it possible toreproduce Macaulay's experiment with a new body of data.

6 Ibid., p. 33.7 W. Braddock Hickman, "The Term Structure of Interest Rates: An Explora-

tory Analysis," National Bureau of Economic Research, 1942, mimeographed.

8 Explanations of theterm structure of interest rates, as interpreted by the Lutz-Miesel-man variant of the expectations hypothesis. For such a comparison,expected yield curves must be determined at one point and actualyield curves at a later point of time. If the expectations hypothesisis valid, Hickman reasoned, then expected yield curves will be corre-lated with observed yield curves.

Hickman found that simply assuming that this year's yield curvewill be the same as next year's gave what he regarded as betterpredictions of subsequently observed yield curves than the expecta-tions hypothesis. This was one of the early uses of an inertia hy-pothesis as a benchmark for evaluating the predictive content of asubstantive hypothesis. Hickman did not employ correlation anal-ysis. If he did, as shall be shown, his conclusion that inertia is thebetter predictor would be more difficult if not impossible to sustain.In addition, he subjected the expectations hypothesis to two addi-tional tests. (These tests, and the data employed are described inAppendix A.) All of his tests are based on the view that the validityof the expectations hypothesis hinges upon accurate forecasts.Meiselman does not regard this finding as relevant. "Anticipationsmay not be realized yet still determine the structure of rates in themanner asserted by the theory." 8

3. CULBERTSON

Culbertson's empirical research is similar to Hickman's; both rantests based on the assumption that forward rates are accurate pre-dictions of future spot rates. Culbertson examined the yields ofshort- and long-term governments for identical periods of time. Heargued that if the expectations hypothesis is valid, then yields toinvestors ought to be the same whether short- or long-term secu-rities are held. (His calculations take into account both incomestreams and capital gains and losses.) He found marked differencesin returns for the same holding periods. Since he found it difficult

8 Meiselman, Term Structure of interest Rates, p. 12. Hickman also had somedoubts about the relevance of his test or any other test. The difficulties in con-ceiving of a means for testing the expectations hypothesis led Conard to con-tend erroneously, as Meiselman's work demonstrates, that only by assuming themarket predicts accurately is it possible ". . .. to build a theory whose predictionscan be meaningfully tested." See Conard, Theory of interest, p. 290.


to believe that speculators would operate in the government secu-rities markets and predict as badly as his results suggeste.1, hejected the expectations hypothesis.9

4. WALKER

Walker's test of the expectations hypothesis also was based on theassumption that the market could predict accurately. However, itwas more like Macaulay's.work in this respect than that of Hick-man and Culbertson. Both he and Macaulay revealed the consist-ency between the implications of accurate expectations and the ex-pectations hypothesis; both observed instances in which theexpectations of the market could be presumed to be accurate; andboth found the behavior of the market was consistent with the ex-pectations hypothesis.1°

Walker's work deals with governmental interest rate policy dur-ing World War II. Around the beginning of that war, the FederalReserve System and the Treasury embarked upon a policy of stabil-izing, through open market operations and the maturity composi-tion of new issues, the existing levels of rates on government se-curities. At that time, the yield curve was sharply rising; the billrate was three-eighths of 1 per cent, one-year securities yielded 1per cent, and long-term securities 2.5 per cent. If the expectationshypothesis is correct, the prestabilization term structure impliedthat future short-term rates were expected to be higher than exist-ing short-term rates. In contrast, the stabilization policy impliedthat future short-term rates would be the same as current short-term rates. When the financial community became convinced thatthe monetary authorities could and would make this policy effec-tive, it also became convinced that existing long-term rates wereinconsistent with revised expectations of future short-term rates:

U". the explanation of broad movements in the term structure of rates mustbe sought principally in factors other than behavior governed by interest rateexpectations." See John M. Culbertson, "The Term Structure of Interest Rates,"Quarterly Journal of Economics, November 1957, p. 502.

Meiselman, Term Structure of Interest Rates, p. 12, regards this and Hick-man's work as tests of nonexistent implications of the expectations hypothesis.

10 Charts E. Walker, "Federal Reserve Policy and the Structure of InterestRates on Government Securities," Quarterly Journal of Econo,nics, February1954, p. 19.

10 Explanations of thelong-term rates were too high. Hence, there was a tremendous shiftout of short- and into long-term securities by the holders of govern-mental obligations. Such a shift is implied by the expectations hy-pothesis, given the prewar term structure and its wartime stabiliza-tjOfl.11 This shift in large part converted the stabilized yield on billsto a nominal rate similar to some other wartime prices.

Walker's results, unlike Macaulay's findings, cannot be inter-preted as providing unambiguous support for the expectations hy-pothesis because they are also consistent with an implication of theliquidity preference hypothesis. Liquidity preference as a theoryof the term structure of interest rates implies that the longer theterm to maturity of a security, the higher its yield. Yield differen-tials between long- and short-term securities constitute equalizingdifferences that reflect differences in risks of capital losses. Theestablishment of a ceiling on long-term bond yields implies a flooror support price for their capital values. A price support program•for long-term bonds implies that much of the risk of capital loss iseliminated. Therefore, long maturities become relatively more at-tractive investment media.

Although Walker's results do not discriminate between expecta-tions and liquidity preference, they do discriminate between ex-pectations and liquidity preference on the one hand and marketsegmentation on the other. If the holdings of governments by themajor institutions of the financial community changed as much asWalker reports they did, this constitutes evidence against themarket segmentation hypothesis; if the market segmentation hy-pothesis is correct, Walker should not have observed a shift in thematurity distribution of governments by the major institutions ofthe financial community.12

11 If a rising yield curve exists, long-term securities yield more than short-term because the market anticipates offsetting losses on capital account attribu•table to holding long-term securities. The elimination of these anticipated capi-tal losses implies that the yield of long-term securities is truly greater than thatof short-term securities.

Conversely a declining yield curve implies that future short-term rates will belower. Hence the holders of long-term securities trade a lower income on currentaccount for anticipated capital gains. The stabilization of such a yield curvemeans that these anticipated capital gains cannot be realized, hence, that theyield of short-term securities is truly greater than that of long-term securities.

12 This interpretation of Walker's findings as well, as the contention that hisresults are consistent with liquidity preference does not appear in the original


The expectations hypothesis has been rejected for its unrealisticassumptions, particularly the assumption that short- and long-termsecurities of equivalent default risk can be treated as perfect sub-stitutes. Many practitioners in financial markets, committing thefallacy of composition, reason that no one regards bills and long-term bonds as alternatives because they observe that many institu-tions specialize in a particular maturity spectrum. As long as someranges of maturities are considered as alternatives by individualparticipants in this market, and in the aggregate these ranges coverthe entire maturity spectrum, the market will act as though billsand bonds are alternatives. Yet every participant in this marketmay deal in a highly circumscribed maturity spectrum.

Mrs. Robinson has contended that the purchasers of a consolmust know the course of future interest rates for . . every dayfrom today till Kingdom Come." 13 Hickman and Luckett haveenunciated, less colorfully, essentially the same argument.'4

Presumably the size of the bonus a promising high school or col-lege baseball player receives in exchange for his affiliation with amajor league club is a function of his expected performance as aball player. This interpretation, which is widely accepted, impliesthat the market predicts the performance of a ball player over hisentire career. In order to properly calculate the size of these. bo-nuses, the market must predict batting averages, fielding perform-ance, and, in the case of pitchers, pitching effectiveness. Emotionalstability, which appears to be irrelevant for determining futureshort-term rates, must also be predicted for ball players, si;rice manybecome emotionally unstable in the face of severe competition andhence lose some of their economic value.15

paper. Walker regarded his evidence as supporting the Lutz variant of expecta-tions. For another statement of what the market segmentation hypothesis is, seeConard, Theory of Interest, p. 304.

13 See Joan Robinson, "The Rate of Interest," Econonzetrica, April 1951, p.102.

14 Dudley G. Luckett, "Professor Lutz and the Structure of Interest Rates,"Quarterly Journal of Economics, February 1959, p. 131. Hawtrey also seems tobe a member of the school that rejects the expectations hypothesis because ofdifficulties in predicting short-term rates. He al-gues that short- and long-termrates are determined in completely segregated and independent markets. SeeRalph G. Hawtrey, "A Rejoinder," The Manchester School, October 1939, p. 156.

15 The objection to the expectations hypothesis for the lack of ":realism" inits assumptions has led to an attempt to find an alternative, more realistic set

12 Explanations of the5. MEISELMAN

Meiselman is the first investigator to employ an operational testof the expectations hypothesis that does not depend upon accurateforesight for its validity. If a relationship exists between expecta-tions and the term structure of interest rates, then its existence canbe detected despite inaccurate predictions. The understanding byeconomists of how expectations are formed and revised in the lightof new information has improved enormously in recent years.Meiselman, by utilizing this knowledge, was able to make the ex-pectations hypothesis operational even when the market could notanticipate future rates of interest correctly. He showed that ex-pectations, whether or not they are correct, nevertheless affect theterm structure of rates. His results constitute striking evidence thatthe expectations hypothesis has empirical validity.'6

The expectations hypothesis implies that the term structure ofinterest rates constitutes at one moment of time a set of predictionsof short-term rates at various moments of time in the fuwre. Forevery instant of time, there exists a term structure or yield curveand a set of implicit forward rates. These forward rates are, if thehypothesis is correct, expected short-term rates. If two term struc-tures separated temporally are compared, the earlier contains pre-dictions of future short-term rates and the later the data, i.e., therealized or actual short-term rates necessary for an evaluation of theaccuracy of these predictions. Recent work on expectations suggeststhat if a realized or actual short-term rate is above its predictedlevel, then the predictions for other rates, yet to be realized, will berevised upward. Conversely, if the actual rate is below the predicted,then other predicted rates will be revised downward during the timeinterval between observations.

of assumptions. Sec Burton G. Malkici, "Expectations, Bond Prices, and theTerm Structure of Interest Rates," Quarterly Journal of Economics, May 1962,No. 2, p. 197. The author claims his model is ". . . in closer conformity with thepractices of bond investors who had always considered the Lutz theory chimeri-cal." (See P. 218.) Conformity here should not be interpreted as predictingbetter; there is no test of the predictive powers of the models in the Malkielpaper. Conformity refers to the conformation of the assumptions of Malkiel'smodel with descriptions of how bond investors behave.

10 Meiselnian, Term Structure of interest Rates, Chapter 2.


To illustrate: Assume at T0, say January 1, 1960, the followingrelationships between yield and term to maturity are revealed bythe market:

Yields as a Function of Term to Maturity at T01-year governments yield 1.0 per cent2 2.03 3.04 4.0

The expectations hypothesis, given this data at T0, implies that themarket expects future one-year rates to be higher than the cur-rent one-year rate. Since the one-year rate is 1 per cent and thetwo-year rate 2 per cent, the forward rate on one-year money oneyear hence must be 3 per cent for the returns on these alternativesto be equal. Analogously, if the current two-year rate is 2 per centand the three-year rate 3 per cent, then the forward rate on one-year money two years later must be high enough to compensate forthe difference between 2 and 3 per cent for two years. Therefore, aone-year rate of 5 per cent is implied for two years hence.

Market Predictions at T0 of Expected One-Year RatesExpected one-year rate for T1, the year beginning 1/1/61, is 3.0 per cent

1/1/62, 5.0T3 1/1/63, 7.0

Assume at T1, a year later, that the following relationships be-tween yield and term to maturity are revealed by the

Yields as a Function, of Term to Maturity at T11-year governments yield 2.0 per cent2 3.33 4.0

Clearly the one-year rate observed in the market at T1 (2 per cent)is less than it was expected to be a year ago (3 per cent). The differ-ence between the anticipated one-year rate one year hence at T0and the realized one-year rate at T1 (both rates are for an identicalmoment of time but are measured one year apart) is defined as theerror. If recently acquired knowledge on the formation of expecta-tions is correct, then forecasts of expected one-year rates for T2 and

1

14 Explanations of theT3, i.e., for January 1, 1962, and 1963, will have been revised down-ward during the year 1960, or between T0 and T1.

One can infer from the term structure of interest rates at T0 andT, how much these estimates of future short-term rates have beenrevised.

Market Predictions at T0 and T,Change in Forecast,

Expected One-Year or Magnitude ofRate for One Year, Forecast Revision

Beginning on T0 T1 (per cent)January 1, 1962 (T2) 5.0 4.6 —0.4January 1, 1963 (T3) 7.0 5.4 — 1.6

At T1 the expected one-year rates beginning at T2 and T3 are 4.6and 5.4 per cent respectively. The difference between 5.0 and 4.6per cent measures the change in the forecast one-year rate for T2;the difference between 7.0 and 5.4 measures the change in the fore-cast one-year rate for T3. Hence, if the expectations hypothesis iscorrect, then errors and forecast changes should be positively cor-related.'7 Meiselman found that his error terms (i.e., the differencebetween predicted and actual one-year rates) and his forecast re-visions were in fact positively correlated.

The distinction between anticipated and unanticipated interestrate changes is crucial for an understanding of how Meiselmantested the expectations hypothesis. If forward rates a year apart areas depicted by Chart 1, then the expectations hypothesis would im-ply that there has been no change in the rates forecast. Yet the ratesfor one-, two-, and three-year maturities must have changed duringthis year; yield curves were not constant. Nevertheless the ex-pected one-year rates for particular moments of time were un-changed. The observations that are correlated, i.e., the error termand the forecast revision, refer to interest rates for particular(lates.18

17 Meiselinan defines the error as the spot minus the forward; the revision ofthe forecast is defined as the later forecast less the earlier.

18 An implication of this distinction is the proposition that stock prices canvary over time with no change iii expectations of future earnings, if the marketexpects earnings to fluctuate. Hence, insofar as investors anticipate cyclicalchanges in the profitability of enterprises, anticipated cyclical variations in stockprices should exist.


Meiselman cot-related errors with contemporaneous revisions inforecasts. For the example used, there are two forecast revisions,—0.4 and — 1.6, that are correlated with the error, — 1.0. The futurespot rates whose estimates were revised will be observed in themarket as spot, and not forward, rates one and two years after thespot rate in the error term can be observed. For the data Meiselmanemployed, the future spot rates whose estimates were revised willbe observed in the market spot rates one through eight yearsafter the spot rate in his error term can be observed. In both theexample and Meiselman's work, forward rates pertaining to subse-quently observable one-year spot rates for particular moments ofcalendar time were observed a year apart. The difference betweenobservations which pertain to the same spot rate are forecast re-

CHART 1Marginal Rates of Interest with Stable Expectations

rates yearly, he observed eight forward rate revisions and. one errorterm every year (with, of course, the exception of the earliest yearthat his data encompasses). Meiselman produced eight :regressionsrelating forward rate revisions to errors observed simultaneously.He found significant relationships for all eight, with correlationcoefficients ranging from a low of .59 to a high of .95. All eight re-gression lines went through the origin, in the sense that the con-stant terms of the regressions were insignificantly different fromzero.

This led to the inference that forward rates are unbiased esti-mates of future spot rates, which implies, when trends in interestrates are ignored, that yield curves are on the average flat. Short-and long-term rates will tend to be equal. if forward rates are

A B

Per cent

1960 1961 1962

visions. Since Meiselman observed

Per cent4

2

0

—----4.

1961 1962 1963

his forward and spot one-year

16 Explanations of thebiased upward, then yield curves, again ignoring trends, are on theaverage positively sloped. Hence, short-term rates will average lessthan long-term rates, and both, on the average, will rise with termto maturity. Such differentials between different terms to maturity,usually referred to as liquidity premiums, reflect the greater liquid-ity of short maturities.'9 Meiselman argues that the absence of aconstant term in his regressions implies the absence of liquiditypremiums. If the constant term is zero, a forward rate that is equalto the subsequently observed actual spot rate, i.e., a zero error term,implies no forecast revision. If forecasts are not revised when theerror term is zero, then Meiselman infers that liquidity premiumsare absent. To show that this inference is incorrect, consider thefollowing formal statement of the hypothesis Meiselman tests:

— = — (1)

Let £ represent expected rates, R spot rates, F forward rates, andL liquidity premiums. The pre-subscript represents a year of cal-endar time. The post-subscript measures the moment a rate is eitherinferred from the term structure or observed as an actual spot rate.The forward and spot rates Meiselman considered were for one yearonly. Hence, t+mEg is. the expected. one-year spot rate for the yeart + m that is inferred from the term structure of interest rates atmoment t. The expected one-year spot rate for the year t + m thatis inferred from the term structure of interest rates at moment£ — 1 is mEti. The difference between the post-subscripts t andI — 1 is, for Meiselman's study, one year.

One cannot observe expected rates directly; the term structure ofinterest rates reveals only forward rates. Whether or not E F, orE + L = F must be established by empirical evidence. Suppose li-quidity premiums exist and they increase monotonically at a de-creasing rate as a function of term to maturity. Then the longer thetime interval between the moment a one-year forward rate is in-

19 The Hicksian view of the term structure of interest rates implies that for-ward rates are biased and high estimates of future short-term rates. He viewedthe "normal" yield curve as being positively sloped. See John R. Hicks, Value

Capital, London 1946, pp. 135—140. Lutz explicitly rejected the view thatliquidity premiums exist because he could observe short-term rates above corre-sponding long-term rates and he regarded this as a contradiction of the liquiditypreference hypothesis. See Lutz, in Tlieoiy of Income Distribution, p. 528.


ferred from a term structure and the moment it becomes a spotrate, the greater the liquidity premium. Similarly, year-to-yearchanges in forward rates for specific calendar years will increase asthey get closer in time to becoming spot rates. The largest increasewill occur during the year a forward rate becomes a spot rate.2°

If the forward rate, F, is e4ual to the expected rate, E, plus aliquidity premium, L, then substituting in (1) yields

(t+mFt t+m1-'t) — (t+mFt_i — t+ns't—_l) = — —

Let — + = AL. Then the restatement of Meiselman'shypothesis becomes

— rnFt_i = — SF11) + — AL.

Letting a = — AL, results in— + (2)

This is the regression equation Meiselman computed. He foundthat the observed constant was insignificantly different from zero.Hence, he inferred that a or — AL is also insignificantly differ-ent from zero.

A zero constant term is equally consistent with either =

AL = 0 or = AL > 0. Hence, this 1)iece of evidence is map-propriate for establishing the validity of the proposition that for-ward rates are unbiased estimates of expected spot rates; it is Con-sistent with the existence of liquidity premiums. The propositionthat forward rates are unbiased estimates of future spot rates re-mains untested.

Meiselman's own work, the work of Hickman, the time series oshort- and long-term governments for the past forty years (to bepresented in Chapter 3), and some new evidence presented here, allsupport the view that the term structure of interest rates, as inter-Feted by the expectations hypothesis, embodies biased and highestimates of future short-term rates. Meiselman used Durand's yieldcurves for high-grade corporates from 1900 through 1954 for histests. For each of these years, Durand estimated a yield curve, if an

20 For the purpose of determining whether or not forward rates arc biased orunbiased estimates of spot rates, the liquidity content of spot rates is irrelevant.It is only the difference, if any, between the liquidity content of forward andspot rates that matters.

8 9 10

SouRcE: 1900—42, Durand, Corporate Bonds; 1943—47, Durand and Winn, BasicYields of Bonds; 1948—51, The Economic Almanac, 1956 (National IndustrialConference Board).

during these fifty-five years, forward rates must have been arith-metically high estimates of spot rates.

If liquidity premiums exist, the. frequency of high estimates oughtto be greater than that of low estimates and the average of the dif-ferences between estimated and actual rates ought to be positive.

18 Explanations of theaverage is computed of the yields for each term to maturity, i.e., anaverage of all fifty-five one-year maturities, two-year maturities, etc.,the composite yield curve which results, reflects average conditionsfor all fifty-five years. This curve is in fact positively sloped (seeChart 2). Since interest rates, if anything, were trending down

CHART 2Average Yield as a Function of Term to Maturity,

Durand Data, 1900—1954

Yield3.55

3.50

3.45

3,40

3.35

3.30

3.25

3.20

Years to maturity


Hence, Meiselman's error terms ought to have a significantly higherfrequency of minus than plus signs and their average ought to benegative. Tests of these implications with the Wilcoxon two-sampleand signed-rank tests lead to their acceptance.21

The foregoing demonstrates that forward one-year rates were onthe average greater than actual one-year rates. It suggests that theywere also greater than expected one-year rates and that they sys-tematically overstate what the market expects one-year rates to be.This conclusion is based on an analysis of the inputs for Meisel-man's independent variable. What about the dependent variable,i.e., the forward-rate changes that are regarded by Meiselman asprediction changes? Since forwarcE rate changes are the differencebetween observations, separated by a year, of forward rates thatpertain to a specific spot rate observable in the future, the firstforward rate must be inferred from data further out on a yieldcurve than the second. Hence, if liquidity preference is operative(if it produces positively sloped yield curves), then the first forwardrate ought to be, on the average, greater than the second. Meisel-man observed prediction changes separated by one through eightyears from the moment of time relevant for the measurement ofthe error term. The first forward rate is, on the average, larger thanthe second for all eight regressions. It is hard to rationalize thisobservation a chance event; the probability of drawing eightsuccessive negative numbers from a population in which negativeand positive numbers are equally represented is less than 1 per cent.On the whole, this evidence is consistent with a positively slopedyield curve that flattens out as term to maturity increases; it iswhat one would expect to be derived from data summarized byChart 2.

Meiselman's changes in forward rates and error terms constitutea measure of the marginal costs, more precisely the rate of changeof yield with respect to term to maturity, of reducing term tomaturity by a year. The pecuniary values at the margin, as revealedby the market, of liquidity changes attributable to changes in term

21 See W. Allen Wallis and Harry V. Roberts, .Slalis!ics: A New Approach,Glencoe, 1956, pp. 596—598. Significance levels of 6 and 2 per cent were producedusing one tail of the normal distribution. Of the fifty-four forward one-yearrates, thirty-five were high and nineteen were low.

20 Explanations of theto maturity of one year are computed. They behave, roughly speak-ing, as one would expect; the longer it takes for a forward rate tobecome a spot rate, the greater the premium of forward over spot.With but two exceptions out of a possible nine cases, liquidity pre-miums decrease monotonically as term tO maturity increases (seeTable I).

TABLE 1

ERROR TERM SAND FORECAST REVISLONSa

YearObse

a One

a Until Secondrvation Becomes—Year Spot Rate Per Cent

error 0 —.143

Mean forward rate revisionC 1

2

3

4

5

6

7

8

—.101—.078—.065—.077—.054—.040—.049—.022

3mese data were obtained through personalwith Mejealman.

bMean of differences between one—year forward and spot rates.

CHean change In one—year forward rates as term to maturitydecreases by one year.

Hickman's data are consistent with Meiselman's findings. Pre-clicted yield curves for the years 1936 through 1942, with a yearbetween the time predicted and actual yield curves are observed,were all high. Even more interesting, and this is consistent withMeiselman's data, Hickman's results show that the longer the inter-val between predicted and observed or actual yield curves, thegreater the bias in the estimates.22 This empirical finding is animplication of a positively sloped yield curve when trends in ratesare absent.

The data (to be presented in Chapter III) on yields of gov-ernments for the nine most recent business cycles, a period ofroughly forty years, clearly indicate that the average yields of

!2 Tlicre arc twenty-cight predictions, all too high. Sec Table A-i. which repro-duces Hickman's data.


short-term governments are less than long-term governments. Allnine cycles, without exception, conform to this generalization.These data constitute additional evidence that the term structureof rates, as interpreted by the expectations hypothesis, yields biasedestimates of future short-term rates. If forward rates are not ex-pected rates, but expected rates plus a liquidity premium, oneshould expect. these time series to show that yields of short-termgovernments are usually less than long-term governments. SinceMeiselman and Hickman worked with Durand's data,. which reflectthe yields of high-grade corpora tes, these data on the relative yieldsof short- and long-term governments for these nine cycles constituteindependent evidence of the existence of bias in the predictions ofthe expectations hypothesis.

Unfortunately, this evidence is not unexceptionable. The fifty-five yearly observations of Durand, which Meiselman used, have adownward trend. In 1900, Durand's basic thirty-year rate was 3.30per cent; in 1954, it was 3.00 per cent. If declining short-term ratesare unanticipated, the predicted rates of the expectations hypothesiswill exceed actual rates. From 1935 through 1942, the downwardtrend is still greater; the thirty-year basic rate fell from 3.50 to 2.65.Hence, if the long-term downward trend in rates has been unantici-pated by the market, the relationship between the yields of short-and long-term governments may be a consequence of forecastingerrors.23

Meiselman, like Walker, produced evidence relevant for eval-uating the validity of the market segmentation hypothesis; unlikeWalker, Meiselman points out the relevance of his work for thishypothesis. .

. the systematic behavior of the yield curve wouldappear to contradict the widely held view that the market for debtclaims is 'segmented' or 'compartmentalized' by maturity and thatrates applicable to specific maturity segments can best be analyzedby rather traditional partial equilibrium supply and demand anal-ysis where transactors act on the basis of preference for specific

23 Hickman found that a snuple 1)FOjCCtiOfl of the previous year's yield curveproduced numerically closer predictions than the expectations hypothesis, whichis consistent with the foregoing interpretation. His finding is also, of course,consistent with an upward bias in the predictions of the expectations hypothesis.

22 Explanations of thematurities. • •

24 The correlation between forward rate revisionsand error terms demonstrates that changes in the yields of one- andtwo-year securities are related to changes in yields of maturities upto nine and ten years. Consequently, at least for this maturityrange, the market is not segmented enough to invalidate this testof the expectations hypothesis.

C. New Evidence

ConfIning tests of the expectations hypothesis to circumstances forwhich expectations can be presumed to be accurate has producedonly fragmentary evidence. Expectations can be presumed to beaccurate only under very special circumstances. Hence, forwardrates can equal expected spot rates and yet differ from realized spotrates. But even this limited approach has not been fully exploited.Clearly, in a world in which spot rates are positive, and this wouldsurely encompass the two most recent decades, one could assumethat the market never expects negative spot rates. Therefore, ifnegative forward rates were observed, this would constitute evi-dence against the expectations hypothesis. Conversely, if negativeforward rates were not observed, this would be evidence for thehypothesis.

The behavior of the term structure of bill yields during Septem-ber 1960 contradicts the expectations hypothesis. In that month theforward rate on one-week money, inferred from the term structureof bill yields with maturities on December 8th and 15th, was oftennegative.25

For nine of the twenty-one trading days in September 1960,negative forward rates for one-week money could be observed. Torestate the foregoing, on these nine dates in September 1960 (andthis same phenomenon could be observed in September 1959) thereexisted some bills whose asked prices were higher than the askedprices for bills with one week less to maturity. Since it is unreason-able to argue that the market expected the spot rate for one-week

24 Meiselman, Term Structure of Interest Rates, p. 34.25 The asked prices reported on the quote sheets of C. J. Devine were the

source of price data. Salomon Bros. and Hutzler quote sheets contained data thatled to the same conclusion.

Term Structure of Interest Rates 23bills on September 8th, or any other week since the end of WorldWar II, to be negative, it follows that forward rates are not ex-pected spot rates.

Critics have rejected the expectations hypothesis because the pre-dictions of future short-term rates implied by the theory differedfrom subsequently observed actual rates. Meiselman argues thatthese critics have rejected the hypothesis for the wrong reasons. Hisposition, that expectations need not be correct to determine theterm structure of interest rates, is, of course, valid. Yet, given freeentry and competition in securities markets, should not one expectto find a relationship between expectations as inferred from theterm structure of interest rates and subsequently observed actualrates? It is of course unreasonable to expect expectations or predic-tions of future short-term rates to be absolutely accurate. New in-formation coming to the market after a prediction is made willlead to prediction revisions and less than perfect forecasts. Yet newinformation should not lead to biases in the estimates; a mean biasshould not be present. Hence, the average difference between pre-clicted and actual rates ought to be insignificantly different fromzero. The absence or presence of a mean bias in the relationshipconstitutes a test of whether or not forward rates are expectedrates. Similarly, for very short intervals between the inference ofpredictions and the observation of actual short-term rates, thereshould be some observable advantage for the expectations hypoth-esis over some form of inertia hypothesis as a predictor of futureshort-rates. If not, why should the market waste its time and energy,which are scarce resources, in trying to predict future short-term1-ates? 20

To control for trends in rates, and to measure forward and actualrates uninfluenced by capital gain considerations, the forward andactual yields of Treasury bills were examined from the beginning of1959 through March 1962. All of the forward rates implicit in the

26 Meiselman went too far in dismissing the work of Hickman and Culbertson.The expectations hypothesis, as he and Lutz interpreted it, does imply that thereought to be equality in the yields of short- and long-term rates in the absenceof trends. If there is not, either the people operating in this market are doingan unbelievably bad job or this constitutes çvidence against the Meiselmansion of the expectations hypothesis.

24 Explanations of the

TABLE 2

OF ERRCRS IN PREDICTING TREASURY BILL RATESa

14—DayRates

28—DayRates

42—DayRates

56—DayRates

63—DayRates

91—DayRates

No. of observations 124 143 146 137 113 125

Frequency of highpredictions 93 132 135 120 91 119

Average size oferrors (per cent) .199 .567 .599 .444 .455 .669

Average actual rates(per cent) 2.34 2.39 2.54 2.67 2.79 2.91

aBI1IS with precisely 182 and 91 days to maturity were used to com-pute the forward 91—day rate. Ninety—one days after this computation,the spot 91—day rate was observed and compared with the forward rate.Similarly, bills with 1261 112, 841 63, 56, 42, 28, and 14 days tomaturity were used to compute forward rates and to measure 8pot rates.

Bid and asked prices1 obtained from government bond dealers, wereaveraged to obtain the prices used. The daily quote sheets of SalornonBros. & Hutzler, C. J. Devine & Co., were the sources of bid and askedprices. These daily price reports quote bid and asked prices of billsfor specified days to maturity from the time payment is received.

Forward 91—day rates were computed by subtracting the current 91—dayrate from twice the current 182—day rate. This method of computing for-ward rates increases the difficulties of detecting an upward bias in theestimates of the expectations hypothesis. It understates forward rela-tive to spot rates. Indeed, if the estimates of the expectations hypo-thesis were unbiased1 this computing procedure would show a downwardbias. Bill yields are bankers discount yields, and equal discount yieldsfor different maturities are not comparable. For example, a 4 per centdiscount yield on a 90—day bill implies a yield on a 360—day basis of4.04 per cent. In contrast, a 4 per cent discount yield on a 180—daybill implies a yield of 4.08 on a 360—day basis. In general, the longerthe term to maturity of a bill, the more its discount yield understatesits bond equivalent yield. Hence, the procedure followed produces lowerestimates of forward rates than would be produced by a correct computation.

term structure of interest rates during that time for two-, four-, six-,eight-, nine-, and thirteen-week bill rates were computed and com-pared with actual yields. The time period under investigation beganand ended with the 91-day bill rate at the same level, approximately2.75 per cent, although it rose sharply to 4.50 per cent and fell to2.25 before it came back to its original level. The results of this in-vestigation are tabulated in Table 2.

These results, along with the evidence already cited, strongly

Term Structure of Interest Rates 25support the belief that forward rates are biased and high estimatesof future short-term rates. Hence they are not the predict ions ofthe market. In addition, these findings support the common beliefthat there exists a preference for short-term over long-term securitiesin the market. This preference produces a yield differential thatconstitutes an equalizing difference. The greater pecuniary yield oflong-term securities represents compensation for the nonpecuniaryadvantages associated with holding short-term securities.

These findings also suggest that the futures market for moneymay be unlike other futures markets. Generally, one finds that for-ward prices are below corresponding SPOt prices when spot pricesare rising and above them when spot prices are falling. For thefutures market for money, however, forward rates in the Treasurybill market are typically above spot rates even when the latter arerising. During an upswing, the extent to which this occurs narrows,and some reversals, i.e., spot rates in excess of forward rates, occur.

these reversals are su prisingly infrequent.On theoretical grounds, one should expect liquidity

to vary with the level of interest rates. Treasury bills, like othersecurities, can be viewed as providing two streams of income: oneis a pecuniary yield measured by interest rates; the other is a non-pecuniary yield as a money substitute. The average difference in28- and 56-clay bill yields can be viewed as an equalizing differencethat reflects the greater value of the former as a money substitute.Economists customarily think of a rise in interest rates as implyingan increase in the cost of holding money. By parity of reasoning,an increase in interest rates should also imply an increase in thecost of holding money substitutes. Since 28-clay bills are bettermoney substitutes than 56-day bills, a rise in interest rates impliesthat the opportunity costs of holding the former should rise relativeto that of holding the latter. For this condition to be satisfied,yields of 56-clay bills must rise relative to those of 28-day bills. Sucha rise implies an increase in liquidity premiums, i.e., an increase inthe spread between forward and actual 28-day rates. This reasoningis consistent with the results obtained for the range of bill maturitiesstudied; the opportunity costs of holding any specified maturity,instead of a longer and hence less liquid maturity, increases as

26 Explanations of theinterest rates rise. Conversely, these opportunity costs decrease whenrates fall. Within the range of bill maturities observed, and con-trary to what is true for the yield curve as a whole, yield curvesare steepest when rates are high and flattest when rates are low.

If the spread between 28- and 56-day bills increases with a rise inrates, and if liquidity premiums increase, then the premium offorward over spot money should also increase. This implies thatwhat Meiselman and Hickman erroneously regarded as error terms,the difference between forward and subsequently observed spotrates, should be a positive function of the current level of spot rates.To determine whether or not this inference is correct, the differencebetween forward and subsequently observed 28-day spot rates wasregressed on current 28-day spot rates. This is equivalent to regress-ing liquidity premiums plus or minus a forecasting error on current28-day rates. These results are consistent with the hypothesis thatliquidity premiums rise with the level of spot rates. The premiumof forward over spot 28-day rates increases by one basis point forevery increase of about five basis points in the spot rate.

The foregoing conclusion was derived from 137. monthly observa-tions during the three business cycles from October 1949 throughFebruary 1961. They are supported by the results obtained from aregression using 138 weekly observations of 91- and 182-clay billsfrom January 1959 through February 1961. For the latter test, theregression coefficient was about twice the former. A rise of about twoand a half basis points in the 91-day bill rate is associated with arise of about one basis point in the premium of forward over spot91-day rates.27

27 For the 91-day bills, the weekly obscrvations cover a period when there were182- and 91-day bills outstanding simultaneously. regression coefficient was.43 with a standard error of .05.

For the 28-day bills, observations were obtained once a month. Typically, morethan one observation could have been used in any month. The observationchosen was the one closest tothe middle of the month. The regression coefficientwas .22 with a standard error of .03.

The effects of bankers discount were eliminated from these data.The association of a rise in liquidity premiums with a rise iii the level of rates

can also be shown by regressing the difference between forward and subsequentlyobserved spot rates upon their sum.

The validity of these tests depends upon the absence of positive correlationbetween forecasting errors and spot rates. Unfortunately it is difficult to disen-tangle forecasting errors from liquidity premiums.

Term Structure of Interest Rates 27Since both interest rates and business conditions vary with the

cycle, the finding that liquidity premiums rise with interest ratesraises the question, are liquidity premiums a function of the levelof interest rates or of the stage of the business cycle? In order toinvestigate this question, forward and actual 28-day bill rates werecomputed monthly from the term structure of 56- and 28-day billsfor the three latest complete business cycles. During these threecycles, there was an upward trend in interest rates. Therefore, ifliquidity premiums vary with the level of rates, it should be pos-sible to observe that they rise secularly. The regression of the dif-ference between predicted and actual 28-day rates on time for thesethree cycles does indicate an upward trend. Hence, liquidity pre-miums are positively related to the level of interest rates.28

The existence of liquidity premiums implies that the expectationshypothesis yields biased and high estimates of future short-termrates. It does not reveal in any direct way whether or not themarket has any power to correctly anticipate subsequently observedspot rates. If liquidity premiums are held constant, if expected andnot forward rates are observed, does a significant relationship existbetween these expected rates and subsequently observed spot rates?

Forward rates for specific periods of calendar time and subse-quently observed spot rates for the same periods were subjected tocorrelation analysis. This corrects, in a very crude way, for bias inthe estimates of future spot rates attributable to liquidity premiums.Forward rates, which can be regarded as market predictions whenadjusted for liquidity premiums, were inferred from the ter:m struc-ture of 182- and 91-day bill rates. (These rates were computedusing an average of bid and asked prices adjusted for bankersdiscount.)

The results of this test indicate that the expectations hypothesisdefinitely does have predictive content. For 138 predictions of 91-day bill rates from the beginning of 1959 through the first quarterof 1962, the expectations hypothesis explained 58 per cent of the ob-served variation. The question remains whether an inertia hypothe-

28 Of 137 predictions of the Lutz variant of the expectations hypothesis, 121were high, five low, and eleven were correct. The effects of bankers discount wereeliminated from these data.

28 Explanations of thesis could do equally well or better. Perhaps the observed correlationcould be attributable to serial correlation in the data.

To determine whether or not the results obtained should be im-puted to correct expectations, two variants of an "inertia hypoth-esis" were considered. One "predicted" 91-day bill rates 91 dayshence by assuming no change. The other extrapolated into thefuture the difference between current 91-day rates and those 91days ago.

The correlations for both variants of the inertia hypothesis testedwere the same; each explained 48 per cent of the observed variation.The expectations hypothesis explained approximately 20 per centmore of the observed variation. During most of the period of obser-vation, from about the middle of 1959 through the middle of 1960,there was a sharp rise and fall in rates. For the remainder of theperiod, interest rates were roughly stable. If the two hypotheses arecompared for the period when rates were highly unstable (thisreduces the number of observations to fifty), then expectationsexplain 48 per cent of the observed variations, whereas the variantsof inertia each explain 30 per cent. The comparative advantage ofthe theory was stronger, as one would expect, when interest rateswere unstable.

Is the observed difference between these correlation coefficientssignificant? Could it have occured as a result of chance? To answerthis question, forward and current spot rates were correlated withsubsequently observed spot rates and the partial correlation coeffi-cients were computed. The addition of current spot rates increasedthe fraction of the observed variation explained from 58 to 59 percent. The partial regression coefficient for expectations was sig-nificant and positwe (the partial regression coefficient was .86, witha standard error of .14). In contrast, the partial regression coeffi-cient for inertia was negative and also significant (the regressioncoefficient was — .31, with a standard error of .18).

These results indicate clearly that the expectations hypothesisdoes have predictive content that cannot be attributed to inertia.However, the negative coefficient for inertia requires explanation.The hypothesis presented here views the forward rate as a functionof expected spot rates a liquidity premium. But liquidity pre-

Term Structure of Interest Rates 29miums are a function of the level of spot rates: when current spotrates are high, the premium over SPOt that is reflected in the for-ward rate is also high, and vice versa. Hence, the larger the spotrate, the larger the number that ought to be deducted from forwardrates to obtain the expected rates of the market. Therefore, the neg-ative coefficient which is observed is consistent with the view thatliquidity premiums exist and vary directly with the level of interestrates, more specifically with spOt rates.

To restate this argument more formally, using symbols alreadydefined:

1. = +

2. = t(tRi.).3. =

4.

5• — = + U.

The data used to evaluate the predictive content of the expecta-tions hypothesis are reproduced in Chart 3. The thick line dep.ictsactual 91-day rates. The thin lines indicate forward rates adjustedand unadjusted for liquidity premiums. The point of origin of thethin lines at the thick line represents the moment a forward rateis inferred; the terminal point of the thin line measures the mag-nitude of the forward rate at the moment when the actual 91-dayrate corresponding to this forward rate can be observed. Liquiditypremiums were measured using the regression equation obtainedby regressing the difference between forward and realized 01-dayrates on current spot rates. These results suggest that within therange o maturities encompasse(l by Treasury bills, expectationsdo influence the term structure of interest rates, the marketforecasts future spot rates with some degree of accuracy. However,to obtain the expectations of the market, liquidity premiums mustbe deducted from forward rates.29

29 The fact that forward rates are usually higher than actual spot rates mayhave led Hickman to abandon the search for a relationship between. them. Aninertia hypothesis could produce numerically closer predictions to spot. rates thanthe expectations hypothesis, yet the latter could produce stronger correlations.It is the strength of the correlations, if one accepts the view that liquiditypremiums exist, that is relevant for evaluating these alternatives, Insofar asliquidity premiums are a constant or linear function of forward rates, they do

5

2

5

3

2

30

CHART 3

Explanations of the

Per cent6

Market Expectations of Future 91-Day Bill RatesForward ratesForward rates adjusted for liquidity premiumsSpot rates

A. First Observations of Continuous Four-Week Periods

4

3

0

Per cent6

1959 1960 1961 1962

B. Second Observations of Continuous Four-Week Periods

4

01959 1960 1961 1962

3

5

31Term Structure of Interest Rates

Forward ratesForward rates adjusted for liquidity premiumsSpot rates

C. Third Observations of Continuous Four-Week PeriodsPer cent6

5

4

2

1

0

Per cent6

1959 1960 1961 1962

D. Fourth Observations of Continuous Four-Week Periods

4

3

2

0 I I

1959 1960 1961 1962

32 Explanations of theThus far, this analysis does not reveal how stable the liquidity

preference function is. Is the relationship between spot rates andliquidity premiums stable enough to permit one to estimate liquid-ity premiums for one business cycle and use these estimates to un-cover successfully the expectations of the market, as distinguishedfrom forward rates, for a second cycle? To answer this question,the regression of the difference between forward and subsequentlyobserved 28-day spot rates upon current 28-day spot rates, for thetwo cycles from October 1949 through April 1958, was used to esti-mate liquidity premiums for the following cycle. Then inertia andexpectations were compared as a means of forecasting subsequentlyobserved spot rates. Expectations was definitely the better predictor.The standard error of estimate was .50 for inertia against .38 for ex-

The partial regression coefficient for inertia was — .07;

for expectations, it was .75. The standard error of the regressioncoefficient was .19 for inertia and .16 for expectations. Multiplecorrelation analysis, using forward rates adjusted for liquiditypremiums, yields results almost identical with those obtained withunadjusted forward rates.30

These results suggested that the data Meiselman employed, whichwere compiled by Durand, should be reexamined to see if forwardrates do predict subsequently observed spot rates. Hence forwardand current spot rates were considered as independent variablesand subsequently observed spot rates as the dependent variablein a multiple regression equation. This involves using the samedata Meiselman used to compute what he regards as an error term.No evidence of successful forecasting was detected; inertia appearedto be the better independent variable.

To utilize more recent data that are qualitatively more corn-

not influence the correlation of forward with spot rates. For the two sets of sevenpairs of observations in Hickman's study, representing one-year forecasts, thecorrelation coefficient for expectations was .725; for inertia, When bothvariables were included in a multiple correlation, neither had a significant par.tial correlation coefficient. Hence no basis is provided by correlation analysis forarguing that one or the other variable explained the observed variation. If oneplots forward rates and the variant of inertia Hickman employed, there is almosta constant difference between them.

30 For the three cycles, 1949 to 1961, the simple correlation coefficients inch-cated that expectations explained 88 per cent of the observed variation whereasinertia, i.e., extrapolating no change, explained 82 per cent.

Term Structure of. Interest Rates 33parable to the data Meiselman utilized, the experiment, performedwith forward and spot three-month Treasury bills was repeatedusing monthly forward and spot one-year governments for 1958through 1961. One- and two-year rates were read the fixed ma-turity yield curve published monthly in the Treasury Bulletin.3'Again forward and current spot rates were treated as indepëndenivariables and subsequently observed spot rates as the dependehtvariable. The result is consistent with that using three- ahdmonth bills and reinforces the view that the market hasto forecast successfully. However, taken by itself it does notstitute quite as convincing evidence of the existence of successfUlforecasting. This is what one would expect; it is harder to for&ästa year into the future than it is to forecast for three months.

If the rationalization of the statistical findings using three-six-month bills is correct, then forward rates should have acoefficient and current one-year rates a negative one. One shoUldalso expect to find that the partial correlation coefficient forpectations would be smaller in the case of one- and two-yearury securities than it was for three- and six-month bills.

These anticipations are in general borne out. The sign of the re-gression coefficient for one-year spot rates is negative. For three-and six-month bills, this regression coefficient is 75 per cerit greaterthan its standard error; for one- and two-year govêrnmêi-its, it is athird larger than its standard error. For three- and six-nionth bills,the regression coefficient for forward rates is positive arid sik timesits standard error; in the case of one- and two-year it

its standardPossibly the most convincing evidence that the hiarket fore-

cast, with modest accuracy, one-year spot rates one yèär into thefuture was obtained through the following experithent: Liquiditypremiums embodied in one-year forward rates for cyclewere estimated from an equation derived from the difteréhcê be-tween forward and subsequently observed spot iegressed oncurrent one-year rates for the 1954—58 cycle. The expeCted tátes ofthe market for the 1958—61 cycle were then obtained

31 I am indebted to H. Irving Forman of the National stà thesemeasurements. They are reproduced, along with the related aiid spotrates, in Appendix Table B!.

34 Explanations of thethe estimated liquidity premiums from forward rates. The meansquare errors in. the implicit forecasts of the market, i.e., the differ-ence between forward rates less liquidity premiums and subse-quently observed spot rates were compared with those generatedby assuming next year's one-year spot rates will be identical withcurrent rates. Although neither independent variable appeared insome absolute sense to yield very good forecasts, it is clear that ex-pectations was significantly better as an independent variable thaninertia. For thirty-five monthly observations, the mean square errorwas 2.09 for inertia, .91 for The elimination of li-quidity premiums contributed importantly to this reduction inerror. Without such adjustment, the mean square error of the for-ward rates was 1.91, only slightly less than that for inertia. Theseresults show that if one is predicting one-year rates one year hence,and the current one-year rate is known, adding the two-year rate toone's knowledge constitutes a valuable piece of information.

Time series of forward and spot one-year rates during the period1958 to 1961 are reproduced as Chart 4. These data, as well as thedata for forward and spot three-month bills, suggest that the marketcan detect spot rates that are abnormally high or low. All of theforward rates are biased estimates. However, if one examines theslopes of the lines connecting current spot rates with forward ratesfor one year into the future, these lines appear flattest when cur-rent spot rates are highest. Hence, if the market can abstract fromliquidity premiums (which produce the bias) then it appears thatthe market can forecast. That is, when rates are high, the marketexpects them to fall, and conversely, as the adjusted forward ratesin the lower part of the chart suggest. This is consistent with theview that the market has some notion of what constitutes a normalrate of interest.

What causes the observed difference between the results usingDurand's data on corporates and the recent data on one- and two-year governments? The evidence provides the basis for highlyspeculative answers at best. Durand's data encompass fifty-five yearsand are yearly observations; the data on governments encompassfive years and are monthly observations. Possibly the market can-not distinguish between cyclically and secularly high and low rates

5


CHART 4Forward and Spot One-Year Rates on Government Securities

Per cent Per cent

5

4

3

2

4

3

2

0

SOURCE: Derived from Treasury yield curves, using one- and two-year rates. Therates in the upper section of the chart are from Table B-I; those in the lowersection, from Tables B-2 and B-S.

1958 1959 1960 1961 1962

1

0

36 Explanations of theof 4nterest. If the market could anticipate cyclical changes betterthan secular changes, there would be an observed difference in fore-

accuracy over one cycle as compared with many cycles.When spot rates are high cyclically, their subsequent change is

different from that when they are high secularly. If theforecasts of the market are the same in either case, studies of theaccuracy of forecasts will lead to different results depending uponthe time period under investigation.

Another avenue for explaining secular and cyclical differences isthe study of the stability of liquidity premiums over time. Beforethe 1930's, judging by Durand's data, liquidity premiums weremuch smaller or possibly nonexistent. There seems to have been astructural change in the economy in this respect since the early

Possibly this can be attributed to the abolition of intereston demand deposits, or perhaps to a change in attitude towardrisk that led to changes in liquidity premiums. In any case, insta-bility of liquidity premiums could account for the observed differ-ence in the secular and cyclical correlations of forward and one-year spot rates.

Still another avenue for explaining these findings is data limita-tions. Durand did not use a criterion such as least squares for hiscurve fitting. He fitted only yield curves that do not have maxi-

or minimums. When his yield curves were not flat through-out, they either increased or decreased monotonically with term

maturity and then flattened out. By definition, Durand couldobserve a yield curve with any other shape. He offers no ex-

planation for this self-imposed constraint.In the postwar period, when short-term rates have been above

long-term rates, yield curves have been hump shaped. These curvesat first rise with term to maturity, reach a maximum, and then falland finally flatten out. It is difficult to believe that this was not alsotrue during some of the fifty-five years encompassed by Durand'sdata. If one examines both the data and the curves fitted, it is clearthat humped yield curves could just as correctly have been fittedsome of the time. Since this was not done, one- and two-year ratesderived from Durand's curves are probably high estimates of trueone- and two-year rates, and are high relative to longer maturities.

Term Structure of Interest Rates 37If one examines the yield curves Durand fitted to data in the

1920's, yield curves for governments and corporates have oppositeslopes for three of these years. Indeed, the data on governmentspresented above show short-term governments yielding, on average,less than long-term governments in the 1920's. Durand's findingson corporates indicate just the opposite.

Another difficulty, ignored by both Hickman and MeiselLman, isthe fact that Durand's yield curves are drawn for coupon bonds.Hence, the Hicksian formula for internal rates of return or yieldto maturity, which implicitly assumes the absence of coupons, isinappropriate for computing forward rates. To compute forwardrates correctly, both coupons and yields to maturity, or internalrates of return, must be known.

If one accepts the view that yield curves were, on average, posi.tively sloped during the fifty-five years Durand observed, then coti-pon rates for bonds with one or two years to maturity must have,on average, exceeded internal rates of return. If coupons exceedinternal rates of return, then it can be shown that the Hicksianformula underestimates forward rates. However, the measurementerrors which can be attributed to ignoring coupons seem to besmall compared to those attributable to uncertainties regardingthe shape of Durand's yield curves. Using coupons of 6 per cent,errors in computing forward rates seem to be on the order of twoor three basis points.

The figures on bill rates collected provide new data to repeatMeiselman's experiments. The results of tests of the expectationshypothesis using Treasury bills are tabulated in Table 3. Treasurybills with terms to maturity of less than six months are the sourceof price data.

Since these correlations are all unambiguously significant, theyprovide additional support for Meiselman's view that a relationshipbetween expectations and the term structure of interest rates exists.His major conclusion—that there is validity in the expectationshypothesis—is sound, despite his failure to isolate unanticipatedchanges in interest rates and to recognize that forward rates werenot expected rates. What about the data Meiselman used? Howare the liquidity premiums related to the level of rates for Durand's

38 Explanations of the

TABLE 3

CORRELAT ION OF FORECAST REVISIONS WI TM ERRORS

AS DEFINED BY

Type of ErrorCorrelationCoefficient

RegressionCoefficient

1. Error in forecast of two-week rates withchanges in expected two—week rates twoweeks hence .37 .40

2. Error in forecast of two-week rates withchanges in expected two—week rates eleven .

weeks hence .36 .26

3. Error in forecast of four—week rateswith changes in expected four—weekrates twelve weeks hence .21 .27

4. Error in forecast of six—week rateswith changes in expected six—week rateseighteen weeks hence .59 .62

5. Error in forecast of eight—week rateswith changes in expected eight—weekrates sixteen weeks hence .85 .59

Source

Line 1: Correlation of changes in predicted two-week bill rates withforecasting errors implied by the expectations hypothesis, i.e., withthe difference between predicted and actual two—week rates. The errorterms were obtained by comparing predictions implied by four— and two—week bill rates with actual two—week bill rates two weeks later. Theprediction changes were obtained from the difference between the pre-dicted two—week rate four weeks hence and then, two weeks later, twoweeks into the future. The first prediction was obtained through theuse of six— and four—week bills; the second was measured through theuse of four— and two—week bills.

Line 2: Correlation of thangea in predicted two—week bill rates as in-ferred from eleven— and nine-week bills and, two weeks later, fromnine— and seven—week bills with the difference between predicted andactual two-week rates. The independent variables for this and the testdescribed in Line 1 are identical.

Line 3: Correlation of changes in predicted four—week bill rateB with theprediction errors implied by the expectations hypothesis. The independ-ent variable is the difference between predictions implied by eight— andfour—week bill rates and, four weeks later, actual four-week bill rates.The dependent variable——the prediction change——is the difference betweenthe predicted four—week rate implied by the sixteen— and twelve—week billrates and, four weeks later, the predicted four—week rate implied by thetwelve— and eight-week bill rates.

Line 4: Correlation of changes in predicted six—week bill rates with pre-diction errors. The independent variable is the difference between pre-dictions implied by twelve— and six—week bill rates and, six weeks later,actual six—week bill rates. The dependent variable, the predictionchange, is the difference between the predicted six—week rate implied

Term Structure of Interest Rates 39NOTES TO TABLE 3 (concluded)

by the twenty—four— and eighteen—week rates and, six weeks later, thepredicted aix—week rate implied by the eighteen— and twelve—week billrates.

Line 5: Correlation of changes in predicted eight—week bill rates withprediction errors. The independent variable is the difference betweenpredictions implied by sixteen— and eight—week bill rates and, eightweeks later, actual eight—week bill, rates. The dependent variable, theprediction change, is the difference between the predicted eight-weekrate implied by the twenty—four— and sixteen—week rates and, eight weekslater, the predicted eight—week rate implied by the sixteen— and eight—week rates. This may be illustrated by the following sample calcula—tion. On November 28, 1961, the sixteen—week rate was 2.61, and theeight—week rate 2.51. The expectations hypothesis implies that theeight—week rate eight weeks hence, on January 23, 1962, is expected tobe 2.71. This is twice the sixteen-week rate less the eight—week rate.The actual eight-week rate on January 23, 1962, eight weeks afterNovember 28, was 2.61. Hence the error is —.10. The first predictionin the data from which Line 5 was derived was inferred from the twenty—four- and sixteen-week rates on November 28, 1961. These were 2.72 and2.61 respectively. Hence the predicted rate for March 20, 1962, whichis three times the twenty—four—week rate less twice the sixteen—weekrate, is 2.94. Eight weeks later, on January 23, 1962, the sixteen—weekrate was 2.72, and the eight-week rate 2.61. Hence the predicted eight—week rate for March 20, 1962, was 2.83, and the prediction change —.11.

aThe existence of liquidity premiums implies that the errors as definedby Maisetman are typically larger than the true errors the market committed.The true errors are the differences between fotward rates minus liquiditypremiums and spot rates; the true forecast revisions are the observedrevisions net of liquidity differences.

data? The regression of the difference between forward and subse-quently observed spot one-year rates against current one-year ratesreveals little variation in the "error" with the level of spot rates.The regression coefficient is .09 with the standard error of .06, andonly about 4 per cent of the variation is explained. In contrast, forthe same regression using forward and spot one-year governmentsfor the 1958—61 cycle, the regression coefficient is one, with a stan-dard error of .10, and 70. per cent of the variation is explained.Clearly the different between forward and spot rates for the gov-ernment data appears to be much more sensitive to variations inspot rates than it is for Durand's data.

The reappearance of a seasonal in the money market in recentyears implies that it is possible to repeat Macaulay's experimentwith a new body of data. if the expectations hypothesis is correct,seasonal adjustment factors ought to vary systematically with• termto maturity. More specifically, just as the time money rates "an-ticipated" seasonal changes in call money rates, changes in, say,

40 Explanations of thesixty-day seasonal adjustment factors ought to "anticipate" changesin thirty-day factors. Hence it should be possible to construct a setof seasonal adjustment factors for sixty-day rates if the factors forthirty-day rates are known; knowledge of seasonal adjustment fac-tors for thirty-day bills implies knowledge of these factors for billsof longer maturity.

To test this hypothesis, weekly moving seasonal adjustment fac-tors were computed for twenty-seven- and fifty-five-day bills for1959, 1960, and 1961, using bid prices unadjusted for bankers dis-count. If the expectations hypothesis is correct, a set of seasonaladjustment factors for fifty-five-day bills constructed out of twenty-seven-day factors ought to be more strongly correlated with actualfifty-five-day factors than just twenty-seven-day factors alone. Forevery week, a simple average of twenty-seven-day factors for thatweek and fOr four weeks in the future was computed. This shouldbe, according to the expectations hypothesis, a fifty-five-day seasonal.The correlation of this set of theoretical seasonal adjustments withactual fifty-five-day adjustment factors was stronger than the cor-relation between twenty-seven- and fifty-five-day factors. Converseresults ought to hold for a fifty-five-day seasonal adjustment con-structed out of twenty-seven-day factors, if the adjustment factorsare obtained by averaging the current twenty-seven-day seasonalwith that of four weeks in the past. This seasonal, when correlatedwith the fifty-five-day seasonal directly computed, ought to exhibitless correlation than exists for the relationship between twenty-seven- and fifty-five-day factors. Hence the rank ordering of cor-

TftBLE 4

COEFFICiENTS 0F CORRELATION BETWEEN WEEKLY SEASONALFACTORS IN TREASURY BILL RATES,

1959—61

Average of 27—DayType of Seasonals (Current andSeasonal 4 Weeks Hence) withProaram 55—Day Seasonal

27—Day Seasonalwith 55—Day

Seasonal

Average of 27—DaySeasonals (Current and

4—Weeks Past) with55—Day Seasonal

Multiplicative .844 .811 .520

Additive .804 .750 .486


relations alone, quite apart from the question of whether or notthere is a significant difference between the correlations, consti-tutes evidence that the market anticipates seasonal movements inrates. These findings are summarized in Table 4.

The Durand data and the data collected for this study provide ameans for discriminating between expectations and liquidity pref-erence on the one hand and market segmentation on the other.The market segmentation hypothesis implies that differences inmaturity account for differences in substitutability between secu-rities. If maturity differences are held constant, then the substi-tutability or the cross elasticity of demand ought also to be con-stant. In contrast, the expectations hypothesis implies that aseven-year security is more like an eight-year security than a one.year security is like a two-year security. The expectations hypothesisimplies that the common element in two securities separated by ayear in maturity increases monotonically as term to maturity in-creases.

Similarly, if one accepts the view that liquidity preference varieswith the level of rates, then the premium increases as the level ofrates increases. Hence, if securities separated by a year in term tomaturity are examined, one should expect the common element toincrease as term to maturity increases. Because both liquidity pref-e'ence and expectations have common implications, this test doesnot discriminate between them. It does, however, produce evidencethat must be regarded. as discriminating between expectations andliquidity preference on the one hand and market segmentation onthe other.

The foregoing tests were Perfolmed with two independent setsof data: the Durand data that Meiselman used and yields to ma-turity, for the latest cycle, read off the yield curve in the TreasuryBulletin by a draftsman. The test employed was a simple rank test.The expectations and liquidity preference hypotheses imply thatthe correlations between securities separated by a year in term tomaturity ought to decrease monotonically as term to maturity in-creases. Hence the theory forecasts a set of ranks that can be com-pared with the observed ranks to see if they are positively corre-lated.

42 Explanations of theConsistent results were obtained using these independent sets

of data. The ranks predicted by the expectations and liquidity pref-erence hypotheses and the actual ranks were highly correlated. Eachset of data consisted of nine pairs of ranks. Using the Olds rankcorrelation test, and interpreting the implications of the liquiditypreference and expectations as implying a one-tail test, both signifi-cance levels were under 2 per cent.32

The foregoing analysis of the implications of liquidity preferenceand expectations for the correlation between the yields of securi-ties separated by a constant time span as term to maturity increasesalso implies that yield curves ought to flatten Out with maturity.Given that the weights assigned to marginal, rates of interest, in thedetermination of average or internal, rates of return, decrease withmaturity, then yield curves must flatten out with maturity. Thisassumes that the variance in forward rates is independent of termto maturity.

The evidence presented supports the Hicksian theory of the termstructure of interest rates; it supports the view that both expecta-tions and liquidity preference determine the term structure of in-terest rates. These results show that forward rates should be inter-preted as expected rates plus a liquidity premium. If forward ratesare so interpreted, then the expectations of the market seem to fore-cast subsequently observed short-maturity spot rates; the relation-ship between expected and subsequently observed spot rates can-not be rationalized as the workings of chance.

With respect to the market segmentation hypothesis, the evidenceis less clear. These findings show that this hypothesis is not of thesame magnitude as liquidity preference and expectations in the de-termination of the term structure of rates. The fact that forwardrates embody short-term forecasts of spot rates that have a per-ceptible degree of accuracy implies that liquidity premiums arestable. Hence the scope for the impact of market segmentationupon the term structure of rates must be limited. The Meiselmanfindings on the relationship between what he termed forecast re-visions and errors support this view, as do the tests presented here.

32 The test employed is described in W. Allen Wallis, "Rough-and-Ready Sta-tistical Tests," Industrial Quality Control, March, 1952.

Term Structure of Interest. Rates 43A proponent of market segmentation may argue that these tests,

in particular, the test based on holding absolute maturity differ-ences constant while varying relative maturity differences, are basedon incorrect interpretations of market segmentation. Economicliterature does not contain a statement of the market segmentationhypothesis that is as rigorous as those available either for liquiditypreference or expectations. Therefore, the possibility of misinter-pretation cannot be easily dismissed. The Walker findings whichdeal with the root of the market segmentation hypothesis are par-ticularly relevant. He showed that institutions have sharply changedthe maturity composition of their holdings in response to marketforces. This seems to strike at the very foundation of the marketsegmentation thesis. The only contrary evidence uncovered—thisis also subject to the same uncertainties about its relevance—is theexistence of negative forward rates in the bill market. Such oc-currences seem to be rare, and therefore relatively insignificant,but should not be dismissed entirely. There is always the possi-bility that more of suchevidence exists or that the effects of marketsegmentation are relatively subtle and the tests employed too crudeto detect its existence.33

38 There were negative forward rates in the bill market in the At thattime rates were relatively low and taxes on bank deposits in Illinois were highenough to make it profitable to take a negative yield rather than be subject totaxation on deposits.

explanations of the term structure of interest rates

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